PAM at IPAM Samy Tindel Universit de Lorraine Rough paths: theory - PowerPoint PPT Presentation

PAM at IPAM Samy Tindel Université de Lorraine Rough paths: theory and applications - Los Angeles 2014 Joint work with Yaozhong Hu, Jingyu Huang and David Nualart Samy T. (Nancy) PAM at IPAM LA 2014 1 / 30

Outline Introduction 1 Motivations Aim of the talk Main results 2 Results for the Stratonovich equation Results for the Skorohod equation Elements of proof 3 Samy T. (Nancy) PAM at IPAM LA 2014 2 / 30

Outline Introduction 1 Motivations Aim of the talk Main results 2 Results for the Stratonovich equation Results for the Skorohod equation Elements of proof 3 Samy T. (Nancy) PAM at IPAM LA 2014 3 / 30

Equation under consideration Equation: Stochastic heat equation on R d : ∂ t u t , x = 1 2 ∆ u t , x + u t , x ˙ W t , x , (1) with t ≥ 0, x ∈ R d . ˙ W general Gaussian noise, with space-time covariance structure. u t , x ˙ W t , x differential: Stratonovich or Skorohod sense. Samy T. (Nancy) PAM at IPAM LA 2014 4 / 30

Outline Introduction 1 Motivations Aim of the talk Main results 2 Results for the Stratonovich equation Results for the Skorohod equation Elements of proof 3 Samy T. (Nancy) PAM at IPAM LA 2014 5 / 30

Resolution of SPDEs More general equation: ∂ t u t , x = Lu t , x + G ( u t , x ) + F ( u t , x ) ˙ W t , x , with (many contributors at the conference) General elliptic operator L Polynomial nonlinearity G Smooth nonlinearity F Links: KPZ equation Filtering, backward equations, stochastic control Question: Can we say more about u in the simple bilinear case u t , x ˙ W t , x ? Samy T. (Nancy) PAM at IPAM LA 2014 6 / 30

Homogenization Problem: Asymptotic regime for t , x = 1 ∂ t u ε 2 ∆ u ε t , x + ε − β u t , x V t ε , εα , x where V stationary random field. Link with SPDEs: Under certain regimes for α, β, V we have u ε → u Analysis through Feynman-Kac formula See Iftimie-Pardoux-Piatnitski, Bal-Gu Samy T. (Nancy) PAM at IPAM LA 2014 7 / 30

Intermittency 2 ∆ u t , x + λ u t , x ˙ Equation: ∂ t u t , x = 1 W t , x Phenomenon: The solution u concentrates its energy in high peaks. Characterization: through moments ֒ → Easy possible definition of intermittency: for all k 1 > k 2 ≥ 1 � � | u t , x | k 1 E lim E [ | u t , x | k 2 ] = ∞ . t →∞ Results: White noise in time: Khoshnevisan, Foondun, Conus, Joseph Fractional noise in time: Balan-Conus Analysis through Feynman-Kac formula Samy T. (Nancy) PAM at IPAM LA 2014 8 / 30

Intemittency: illustration (by Daniel Conus) Simulations: for λ = 0 . 1, 0 . 5, 1 and 2. u(t,x) u(t,x) x x t t u(t,x) u(t,x) x x t t Samy T. (Nancy) PAM at IPAM LA 2014 9 / 30

Polymer measure Independent Wiener measure: d -dimensional Brownian motion B x , Wiener measure P B . � t Hamiltonian for t > 0: − H t ( B x ) = 0 W ( ds , B x s ) . Gibbs polymer measure: for β > 0, t ( B ) = e − β H t ( B x ) dG x d P B . u t , x Studies in the continuous case: Rovira-T, Lacoin, Alberts-Khanin-Quastel. Counterpart of intermittency: Localization. ֒ → See Carmona-Hu, König-Lacoin-Mörters-Sidorova Samy T. (Nancy) PAM at IPAM LA 2014 10 / 30

Localization: illustration 1 (by Frédéric Cérou) Figure : Simple random walk distribution Samy T. (Nancy) PAM at IPAM LA 2014 11 / 30

Localization: illustration 2 (by Frédéric Cérou) Figure : Distribution of the directed polymer in strong disorder regime Samy T. (Nancy) PAM at IPAM LA 2014 12 / 30

Outline Introduction 1 Motivations Aim of the talk Main results 2 Results for the Stratonovich equation Results for the Skorohod equation Elements of proof 3 Samy T. (Nancy) PAM at IPAM LA 2014 13 / 30

Aim of the talk Equation: Stochastic heat equation on R d : ∂ t u t , x = 1 2 ∆ u t , x + u t , x ˙ W t , x , Main issues: for a general Gaussian noise, Resolution for Skorohod and Stratonovich equations. Feynman-Kac representation. Links between Feyman-Kac and pathwise (rough paths) solution. Intermittency estimates. Samy T. (Nancy) PAM at IPAM LA 2014 14 / 30

Outline Introduction 1 Motivations Aim of the talk Main results 2 Results for the Stratonovich equation Results for the Skorohod equation Elements of proof 3 Samy T. (Nancy) PAM at IPAM LA 2014 15 / 30

Description of the noise Encoding of the noise as a random distribution: W = { W ( ϕ ); ϕ ∈ H} centered Gaussian family E [ W ( ϕ ) W ( ψ )] = � ϕ, ψ � H with: � � ϕ, ψ � H = + × R 2 d ϕ ( s , x ) ψ ( t , y ) γ ( s − t ) Λ( x − y ) dx dy ds dt R 2 � = + × R d F ϕ ( s , ξ ) F ψ ( t , ξ ) γ ( s − t ) µ ( d ξ ) ds dt , R 2 γ , Λ positive definite functions. µ tempered measure. Remark: This is standard setting (Peszat-Zabczyk, Dalang). Samy T. (Nancy) PAM at IPAM LA 2014 16 / 30

Outline Introduction 1 Motivations Aim of the talk Main results 2 Results for the Stratonovich equation Results for the Skorohod equation Elements of proof 3 Samy T. (Nancy) PAM at IPAM LA 2014 17 / 30

Stratonovich setting Hypothesis on γ : The function γ satisfies 0 ≤ γ ( t ) ≤ C β | t | − β , with β ∈ ( 0 , 1 ) . Hypothesis on µ : We assume the following integrability condition, � µ ( d ξ ) 1 + | ξ | 2 − 2 β < ∞ . R d Example: Riesz kernel in space, namely Λ( x ) = | x | − η . 0 < η < 2 − 2 β . 0 ≤ γ ( t ) ≤ C β | t | − β . Samy T. (Nancy) PAM at IPAM LA 2014 18 / 30

Feynman-Kac solution Theorem 1. Assume: Previous assumptions on γ and µ , and u 0 ∈ C b ( R d ) . For a Brownian motion B independent of W , set � t � � t ) e V t , x � R d δ 0 ( B x u F u 0 ( B x V t , x = t − r − y ) W ( dr , dy ) , t , x = E B 0 Then u F well-defined and solves the equation: � t � u t , x = p t u 0 ( x ) + R d p t − s ( x − y ) u s , y W ( ds , dy ) , 0 interpreted in the Malliavin - Stratonovich sense. Proof: ֒ → Exponential integrability of V t , x . Samy T. (Nancy) PAM at IPAM LA 2014 19 / 30

Pathwise solution Theorem 2. Assume the previous hypothesis, plus: � µ ( d ξ ) 1 + | ξ | 2 − 2 β − ε < ∞ . R d Consider the equation: � t � u t , x = p t u 0 ( x ) + R d p t − s ( x − y ) u s , y W ( ds , dy ) , (2) 0 interpreted in the Young sense. Then: β 2 ([ 0 , T ]; B 1 − β ) . Eq. (2) admits a unique solution in C B 1 − β is a weighted Besov space on R d . The unique solution to (2) is u F . Samy T. (Nancy) PAM at IPAM LA 2014 20 / 30

Moments estimates Theorem 3. Suppose: c 0 | t | − β ≤ γ ( t ) ≤ C 0 | t | − β . c 1 | x | − η ≤ Λ( x ) ≤ C 1 | x | − η . Then, whenever they are defined, both u ⋄ and u F satisfy: � � � � � � 4 − 2 β − η 4 − η 4 − 2 β − η 4 − η 2 − η k u k 2 − η k exp c 2 t ≤ E ≤ exp C 2 t . 2 − η 2 − η t , x Remarks: (i) This result implies intermittency. (ii) Extensions: other kind of Λ including δ 0 , time independent case. (iii) Proof: Feynman-Kac representation, small ball estimates. Samy T. (Nancy) PAM at IPAM LA 2014 21 / 30

Outline Introduction 1 Motivations Aim of the talk Main results 2 Results for the Stratonovich equation Results for the Skorohod equation Elements of proof 3 Samy T. (Nancy) PAM at IPAM LA 2014 22 / 30

Skorohod setting Hypothesis on γ : The function γ lies in L 1 loc . Hypothesis on µ : We assume the following integrability condition, � µ ( d ξ ) 1 + | ξ | 2 < ∞ . R d Example 1: Riesz kernel in space, namely Λ( x ) = | x | − η . µ ( d ξ ) = c η, d | ξ | − ( d − η ) d ξ . 0 < η < 2. Example 2: White noise in dimension 1. Samy T. (Nancy) PAM at IPAM LA 2014 23 / 30

Skorohod: existence and uniqueness Theorem 4. Assume: Previous assumptions on γ and µ . u 0 ∈ C b ( R d ) . Then Skorohod equation: � t � u t , x = p t u 0 ( x ) + R d p t − s ( x − y ) u s , y δ W s , y 0 admits a unique solution. Proof: ֒ → Wiener chaos expansions and estimates. Samy T. (Nancy) PAM at IPAM LA 2014 24 / 30

Skorohod: representation of moments Theorem 5. Assume: Previous assumptions on γ and µ , and u 0 ≡ 1. u 0 ∈ C b ( R d ) . Then the solution u ⋄ of the Skorohod equation satisfies:     � t � t � t , x ) k � � ( u ⋄ 0 γ ( s − r )Λ( B i s − B j  exp    E = E B r ) dsdr 0 1 ≤ i < j ≤ k for a family of i.i.d Brownian motions in R d . Proof: ֒ → Feynman-Kac formula for approximations. Samy T. (Nancy) PAM at IPAM LA 2014 25 / 30

Outline Introduction 1 Motivations Aim of the talk Main results 2 Results for the Stratonovich equation Results for the Skorohod equation Elements of proof 3 Samy T. (Nancy) PAM at IPAM LA 2014 26 / 30

Feynman-Kac functional Proposition 6. Suppose γ and µ satisfy (with β ∈ ( 0 , 1 ) ): � µ ( d ξ ) 0 ≤ γ ( t ) ≤ C β | t | − β , and 1 + | ξ | 2 − 2 β < ∞ . R d Set: � t � R d δ 0 ( B x V t , x = t − r − y ) W ( dr , dy ) , 0 Then for any λ ∈ R and T > 0: t ∈ [ 0 , T ] , x ∈ R d E [ exp ( λ V t , x )] < ∞ . sup Samy T. (Nancy) PAM at IPAM LA 2014 27 / 30